Properties

Label 5225.2.a.y.1.8
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.692644\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.692644 q^{2} +1.14700 q^{3} -1.52024 q^{4} +0.794464 q^{6} +3.60442 q^{7} -2.43828 q^{8} -1.68438 q^{9} +O(q^{10})\) \(q+0.692644 q^{2} +1.14700 q^{3} -1.52024 q^{4} +0.794464 q^{6} +3.60442 q^{7} -2.43828 q^{8} -1.68438 q^{9} +1.00000 q^{11} -1.74372 q^{12} +7.02467 q^{13} +2.49658 q^{14} +1.35163 q^{16} +0.385149 q^{17} -1.16668 q^{18} +1.00000 q^{19} +4.13428 q^{21} +0.692644 q^{22} -4.88630 q^{23} -2.79671 q^{24} +4.86559 q^{26} -5.37300 q^{27} -5.47960 q^{28} -3.20619 q^{29} +4.99714 q^{31} +5.81275 q^{32} +1.14700 q^{33} +0.266771 q^{34} +2.56068 q^{36} -0.661585 q^{37} +0.692644 q^{38} +8.05731 q^{39} +3.92936 q^{41} +2.86358 q^{42} +1.76009 q^{43} -1.52024 q^{44} -3.38446 q^{46} -2.20788 q^{47} +1.55033 q^{48} +5.99184 q^{49} +0.441767 q^{51} -10.6792 q^{52} +4.15602 q^{53} -3.72158 q^{54} -8.78857 q^{56} +1.14700 q^{57} -2.22074 q^{58} +11.1047 q^{59} +7.70215 q^{61} +3.46124 q^{62} -6.07123 q^{63} +1.32290 q^{64} +0.794464 q^{66} -1.12280 q^{67} -0.585521 q^{68} -5.60460 q^{69} +13.3284 q^{71} +4.10699 q^{72} +5.81274 q^{73} -0.458242 q^{74} -1.52024 q^{76} +3.60442 q^{77} +5.58085 q^{78} -11.1857 q^{79} -1.10969 q^{81} +2.72165 q^{82} -6.08541 q^{83} -6.28512 q^{84} +1.21912 q^{86} -3.67750 q^{87} -2.43828 q^{88} -10.2064 q^{89} +25.3199 q^{91} +7.42837 q^{92} +5.73173 q^{93} -1.52927 q^{94} +6.66724 q^{96} -10.4502 q^{97} +4.15021 q^{98} -1.68438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23} + q^{24} + 3 q^{26} + q^{27} + 46 q^{28} + 9 q^{29} + 14 q^{31} + 18 q^{32} + 4 q^{33} - 13 q^{34} + 12 q^{36} + 9 q^{37} + 5 q^{38} - 22 q^{39} + 4 q^{41} - 6 q^{42} + 28 q^{43} + 17 q^{44} + 27 q^{46} + 14 q^{47} - 4 q^{48} + 32 q^{49} - 40 q^{51} + 14 q^{52} + 3 q^{53} - 39 q^{54} + 34 q^{56} + 4 q^{57} + 26 q^{58} + q^{59} + 2 q^{61} - 3 q^{62} + 45 q^{63} + 5 q^{64} - q^{66} + 37 q^{67} + 26 q^{68} - 7 q^{69} - 7 q^{71} + 16 q^{72} + 42 q^{73} - 43 q^{74} + 17 q^{76} + 21 q^{77} - 64 q^{78} - 10 q^{79} + 31 q^{81} + 22 q^{82} + 14 q^{83} - 32 q^{84} + 37 q^{86} + 29 q^{87} + 9 q^{88} + 15 q^{89} - 22 q^{91} + 26 q^{92} - 18 q^{93} - 44 q^{94} + 71 q^{96} + 8 q^{97} - 10 q^{98} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.692644 0.489773 0.244886 0.969552i \(-0.421249\pi\)
0.244886 + 0.969552i \(0.421249\pi\)
\(3\) 1.14700 0.662222 0.331111 0.943592i \(-0.392576\pi\)
0.331111 + 0.943592i \(0.392576\pi\)
\(4\) −1.52024 −0.760122
\(5\) 0 0
\(6\) 0.794464 0.324339
\(7\) 3.60442 1.36234 0.681171 0.732124i \(-0.261471\pi\)
0.681171 + 0.732124i \(0.261471\pi\)
\(8\) −2.43828 −0.862060
\(9\) −1.68438 −0.561462
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −1.74372 −0.503370
\(13\) 7.02467 1.94829 0.974146 0.225919i \(-0.0725384\pi\)
0.974146 + 0.225919i \(0.0725384\pi\)
\(14\) 2.49658 0.667239
\(15\) 0 0
\(16\) 1.35163 0.337909
\(17\) 0.385149 0.0934123 0.0467062 0.998909i \(-0.485128\pi\)
0.0467062 + 0.998909i \(0.485128\pi\)
\(18\) −1.16668 −0.274989
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.13428 0.902174
\(22\) 0.692644 0.147672
\(23\) −4.88630 −1.01886 −0.509432 0.860511i \(-0.670144\pi\)
−0.509432 + 0.860511i \(0.670144\pi\)
\(24\) −2.79671 −0.570876
\(25\) 0 0
\(26\) 4.86559 0.954221
\(27\) −5.37300 −1.03403
\(28\) −5.47960 −1.03555
\(29\) −3.20619 −0.595374 −0.297687 0.954664i \(-0.596215\pi\)
−0.297687 + 0.954664i \(0.596215\pi\)
\(30\) 0 0
\(31\) 4.99714 0.897513 0.448756 0.893654i \(-0.351867\pi\)
0.448756 + 0.893654i \(0.351867\pi\)
\(32\) 5.81275 1.02756
\(33\) 1.14700 0.199668
\(34\) 0.266771 0.0457508
\(35\) 0 0
\(36\) 2.56068 0.426780
\(37\) −0.661585 −0.108764 −0.0543819 0.998520i \(-0.517319\pi\)
−0.0543819 + 0.998520i \(0.517319\pi\)
\(38\) 0.692644 0.112362
\(39\) 8.05731 1.29020
\(40\) 0 0
\(41\) 3.92936 0.613663 0.306832 0.951764i \(-0.400731\pi\)
0.306832 + 0.951764i \(0.400731\pi\)
\(42\) 2.86358 0.441860
\(43\) 1.76009 0.268412 0.134206 0.990953i \(-0.457152\pi\)
0.134206 + 0.990953i \(0.457152\pi\)
\(44\) −1.52024 −0.229186
\(45\) 0 0
\(46\) −3.38446 −0.499012
\(47\) −2.20788 −0.322052 −0.161026 0.986950i \(-0.551480\pi\)
−0.161026 + 0.986950i \(0.551480\pi\)
\(48\) 1.55033 0.223771
\(49\) 5.99184 0.855978
\(50\) 0 0
\(51\) 0.441767 0.0618597
\(52\) −10.6792 −1.48094
\(53\) 4.15602 0.570873 0.285437 0.958398i \(-0.407861\pi\)
0.285437 + 0.958398i \(0.407861\pi\)
\(54\) −3.72158 −0.506442
\(55\) 0 0
\(56\) −8.78857 −1.17442
\(57\) 1.14700 0.151924
\(58\) −2.22074 −0.291598
\(59\) 11.1047 1.44571 0.722857 0.690998i \(-0.242829\pi\)
0.722857 + 0.690998i \(0.242829\pi\)
\(60\) 0 0
\(61\) 7.70215 0.986160 0.493080 0.869984i \(-0.335871\pi\)
0.493080 + 0.869984i \(0.335871\pi\)
\(62\) 3.46124 0.439578
\(63\) −6.07123 −0.764903
\(64\) 1.32290 0.165362
\(65\) 0 0
\(66\) 0.794464 0.0977918
\(67\) −1.12280 −0.137172 −0.0685859 0.997645i \(-0.521849\pi\)
−0.0685859 + 0.997645i \(0.521849\pi\)
\(68\) −0.585521 −0.0710048
\(69\) −5.60460 −0.674714
\(70\) 0 0
\(71\) 13.3284 1.58179 0.790893 0.611955i \(-0.209617\pi\)
0.790893 + 0.611955i \(0.209617\pi\)
\(72\) 4.10699 0.484014
\(73\) 5.81274 0.680330 0.340165 0.940366i \(-0.389517\pi\)
0.340165 + 0.940366i \(0.389517\pi\)
\(74\) −0.458242 −0.0532696
\(75\) 0 0
\(76\) −1.52024 −0.174384
\(77\) 3.60442 0.410762
\(78\) 5.58085 0.631906
\(79\) −11.1857 −1.25849 −0.629246 0.777207i \(-0.716636\pi\)
−0.629246 + 0.777207i \(0.716636\pi\)
\(80\) 0 0
\(81\) −1.10969 −0.123299
\(82\) 2.72165 0.300556
\(83\) −6.08541 −0.667961 −0.333980 0.942580i \(-0.608392\pi\)
−0.333980 + 0.942580i \(0.608392\pi\)
\(84\) −6.28512 −0.685762
\(85\) 0 0
\(86\) 1.21912 0.131461
\(87\) −3.67750 −0.394270
\(88\) −2.43828 −0.259921
\(89\) −10.2064 −1.08187 −0.540936 0.841064i \(-0.681930\pi\)
−0.540936 + 0.841064i \(0.681930\pi\)
\(90\) 0 0
\(91\) 25.3199 2.65424
\(92\) 7.42837 0.774461
\(93\) 5.73173 0.594353
\(94\) −1.52927 −0.157732
\(95\) 0 0
\(96\) 6.66724 0.680472
\(97\) −10.4502 −1.06106 −0.530529 0.847667i \(-0.678007\pi\)
−0.530529 + 0.847667i \(0.678007\pi\)
\(98\) 4.15021 0.419235
\(99\) −1.68438 −0.169287
\(100\) 0 0
\(101\) −12.1058 −1.20457 −0.602285 0.798281i \(-0.705743\pi\)
−0.602285 + 0.798281i \(0.705743\pi\)
\(102\) 0.305987 0.0302972
\(103\) −11.9174 −1.17426 −0.587129 0.809494i \(-0.699742\pi\)
−0.587129 + 0.809494i \(0.699742\pi\)
\(104\) −17.1281 −1.67955
\(105\) 0 0
\(106\) 2.87864 0.279598
\(107\) 9.16324 0.885844 0.442922 0.896560i \(-0.353942\pi\)
0.442922 + 0.896560i \(0.353942\pi\)
\(108\) 8.16828 0.785993
\(109\) 9.05838 0.867635 0.433817 0.901001i \(-0.357166\pi\)
0.433817 + 0.901001i \(0.357166\pi\)
\(110\) 0 0
\(111\) −0.758839 −0.0720258
\(112\) 4.87186 0.460347
\(113\) 0.699650 0.0658176 0.0329088 0.999458i \(-0.489523\pi\)
0.0329088 + 0.999458i \(0.489523\pi\)
\(114\) 0.794464 0.0744084
\(115\) 0 0
\(116\) 4.87419 0.452557
\(117\) −11.8322 −1.09389
\(118\) 7.69162 0.708071
\(119\) 1.38824 0.127260
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 5.33485 0.482994
\(123\) 4.50699 0.406381
\(124\) −7.59688 −0.682220
\(125\) 0 0
\(126\) −4.20520 −0.374629
\(127\) 18.6133 1.65167 0.825834 0.563914i \(-0.190705\pi\)
0.825834 + 0.563914i \(0.190705\pi\)
\(128\) −10.7092 −0.946569
\(129\) 2.01883 0.177748
\(130\) 0 0
\(131\) −8.82334 −0.770899 −0.385450 0.922729i \(-0.625954\pi\)
−0.385450 + 0.922729i \(0.625954\pi\)
\(132\) −1.74372 −0.151772
\(133\) 3.60442 0.312543
\(134\) −0.777700 −0.0671831
\(135\) 0 0
\(136\) −0.939099 −0.0805271
\(137\) −3.11018 −0.265720 −0.132860 0.991135i \(-0.542416\pi\)
−0.132860 + 0.991135i \(0.542416\pi\)
\(138\) −3.88199 −0.330457
\(139\) 7.07826 0.600370 0.300185 0.953881i \(-0.402952\pi\)
0.300185 + 0.953881i \(0.402952\pi\)
\(140\) 0 0
\(141\) −2.53244 −0.213270
\(142\) 9.23180 0.774716
\(143\) 7.02467 0.587432
\(144\) −2.27667 −0.189723
\(145\) 0 0
\(146\) 4.02616 0.333207
\(147\) 6.87266 0.566848
\(148\) 1.00577 0.0826738
\(149\) 2.35944 0.193293 0.0966466 0.995319i \(-0.469188\pi\)
0.0966466 + 0.995319i \(0.469188\pi\)
\(150\) 0 0
\(151\) 3.09275 0.251684 0.125842 0.992050i \(-0.459837\pi\)
0.125842 + 0.992050i \(0.459837\pi\)
\(152\) −2.43828 −0.197770
\(153\) −0.648739 −0.0524474
\(154\) 2.49658 0.201180
\(155\) 0 0
\(156\) −12.2491 −0.980712
\(157\) 10.8156 0.863179 0.431589 0.902070i \(-0.357953\pi\)
0.431589 + 0.902070i \(0.357953\pi\)
\(158\) −7.74771 −0.616375
\(159\) 4.76697 0.378045
\(160\) 0 0
\(161\) −17.6123 −1.38804
\(162\) −0.768622 −0.0603886
\(163\) 3.37074 0.264017 0.132009 0.991249i \(-0.457857\pi\)
0.132009 + 0.991249i \(0.457857\pi\)
\(164\) −5.97359 −0.466459
\(165\) 0 0
\(166\) −4.21502 −0.327149
\(167\) 8.37434 0.648026 0.324013 0.946053i \(-0.394968\pi\)
0.324013 + 0.946053i \(0.394968\pi\)
\(168\) −10.0805 −0.777728
\(169\) 36.3460 2.79584
\(170\) 0 0
\(171\) −1.68438 −0.128808
\(172\) −2.67577 −0.204026
\(173\) 8.91885 0.678088 0.339044 0.940771i \(-0.389897\pi\)
0.339044 + 0.940771i \(0.389897\pi\)
\(174\) −2.54720 −0.193103
\(175\) 0 0
\(176\) 1.35163 0.101883
\(177\) 12.7372 0.957384
\(178\) −7.06936 −0.529871
\(179\) 8.35181 0.624243 0.312122 0.950042i \(-0.398960\pi\)
0.312122 + 0.950042i \(0.398960\pi\)
\(180\) 0 0
\(181\) 23.1394 1.71994 0.859968 0.510349i \(-0.170484\pi\)
0.859968 + 0.510349i \(0.170484\pi\)
\(182\) 17.5376 1.29998
\(183\) 8.83439 0.653057
\(184\) 11.9141 0.878322
\(185\) 0 0
\(186\) 3.97005 0.291098
\(187\) 0.385149 0.0281649
\(188\) 3.35651 0.244799
\(189\) −19.3666 −1.40871
\(190\) 0 0
\(191\) 11.4744 0.830262 0.415131 0.909762i \(-0.363736\pi\)
0.415131 + 0.909762i \(0.363736\pi\)
\(192\) 1.51737 0.109506
\(193\) −17.3482 −1.24875 −0.624374 0.781125i \(-0.714646\pi\)
−0.624374 + 0.781125i \(0.714646\pi\)
\(194\) −7.23827 −0.519678
\(195\) 0 0
\(196\) −9.10907 −0.650648
\(197\) 13.6075 0.969495 0.484748 0.874654i \(-0.338911\pi\)
0.484748 + 0.874654i \(0.338911\pi\)
\(198\) −1.16668 −0.0829122
\(199\) 8.75803 0.620840 0.310420 0.950599i \(-0.399530\pi\)
0.310420 + 0.950599i \(0.399530\pi\)
\(200\) 0 0
\(201\) −1.28785 −0.0908383
\(202\) −8.38499 −0.589966
\(203\) −11.5564 −0.811103
\(204\) −0.671594 −0.0470210
\(205\) 0 0
\(206\) −8.25452 −0.575119
\(207\) 8.23040 0.572053
\(208\) 9.49478 0.658345
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 20.2964 1.39726 0.698631 0.715482i \(-0.253793\pi\)
0.698631 + 0.715482i \(0.253793\pi\)
\(212\) −6.31817 −0.433933
\(213\) 15.2877 1.04749
\(214\) 6.34686 0.433862
\(215\) 0 0
\(216\) 13.1009 0.891400
\(217\) 18.0118 1.22272
\(218\) 6.27423 0.424944
\(219\) 6.66723 0.450530
\(220\) 0 0
\(221\) 2.70554 0.181994
\(222\) −0.525605 −0.0352763
\(223\) 19.8774 1.33109 0.665546 0.746357i \(-0.268199\pi\)
0.665546 + 0.746357i \(0.268199\pi\)
\(224\) 20.9516 1.39989
\(225\) 0 0
\(226\) 0.484608 0.0322357
\(227\) −0.523563 −0.0347501 −0.0173751 0.999849i \(-0.505531\pi\)
−0.0173751 + 0.999849i \(0.505531\pi\)
\(228\) −1.74372 −0.115481
\(229\) 18.6726 1.23392 0.616960 0.786995i \(-0.288364\pi\)
0.616960 + 0.786995i \(0.288364\pi\)
\(230\) 0 0
\(231\) 4.13428 0.272016
\(232\) 7.81757 0.513248
\(233\) −1.69912 −0.111313 −0.0556565 0.998450i \(-0.517725\pi\)
−0.0556565 + 0.998450i \(0.517725\pi\)
\(234\) −8.19553 −0.535758
\(235\) 0 0
\(236\) −16.8819 −1.09892
\(237\) −12.8300 −0.833401
\(238\) 0.961554 0.0623283
\(239\) −29.7115 −1.92188 −0.960938 0.276763i \(-0.910738\pi\)
−0.960938 + 0.276763i \(0.910738\pi\)
\(240\) 0 0
\(241\) −16.6028 −1.06948 −0.534741 0.845016i \(-0.679591\pi\)
−0.534741 + 0.845016i \(0.679591\pi\)
\(242\) 0.692644 0.0445248
\(243\) 14.8462 0.952383
\(244\) −11.7092 −0.749602
\(245\) 0 0
\(246\) 3.12174 0.199035
\(247\) 7.02467 0.446969
\(248\) −12.1844 −0.773710
\(249\) −6.97998 −0.442339
\(250\) 0 0
\(251\) 13.7486 0.867807 0.433903 0.900959i \(-0.357136\pi\)
0.433903 + 0.900959i \(0.357136\pi\)
\(252\) 9.22976 0.581420
\(253\) −4.88630 −0.307199
\(254\) 12.8924 0.808942
\(255\) 0 0
\(256\) −10.0635 −0.628966
\(257\) 26.7962 1.67150 0.835752 0.549108i \(-0.185032\pi\)
0.835752 + 0.549108i \(0.185032\pi\)
\(258\) 1.39833 0.0870563
\(259\) −2.38463 −0.148174
\(260\) 0 0
\(261\) 5.40045 0.334280
\(262\) −6.11143 −0.377566
\(263\) −17.2355 −1.06279 −0.531395 0.847124i \(-0.678332\pi\)
−0.531395 + 0.847124i \(0.678332\pi\)
\(264\) −2.79671 −0.172125
\(265\) 0 0
\(266\) 2.49658 0.153075
\(267\) −11.7067 −0.716439
\(268\) 1.70693 0.104267
\(269\) −17.9313 −1.09329 −0.546646 0.837364i \(-0.684096\pi\)
−0.546646 + 0.837364i \(0.684096\pi\)
\(270\) 0 0
\(271\) −9.75162 −0.592369 −0.296184 0.955131i \(-0.595714\pi\)
−0.296184 + 0.955131i \(0.595714\pi\)
\(272\) 0.520580 0.0315648
\(273\) 29.0419 1.75770
\(274\) −2.15424 −0.130143
\(275\) 0 0
\(276\) 8.52036 0.512865
\(277\) −20.9151 −1.25667 −0.628334 0.777943i \(-0.716263\pi\)
−0.628334 + 0.777943i \(0.716263\pi\)
\(278\) 4.90271 0.294045
\(279\) −8.41711 −0.503919
\(280\) 0 0
\(281\) 1.52898 0.0912115 0.0456058 0.998960i \(-0.485478\pi\)
0.0456058 + 0.998960i \(0.485478\pi\)
\(282\) −1.75408 −0.104454
\(283\) 7.92671 0.471194 0.235597 0.971851i \(-0.424295\pi\)
0.235597 + 0.971851i \(0.424295\pi\)
\(284\) −20.2624 −1.20235
\(285\) 0 0
\(286\) 4.86559 0.287708
\(287\) 14.1631 0.836019
\(288\) −9.79091 −0.576935
\(289\) −16.8517 −0.991274
\(290\) 0 0
\(291\) −11.9864 −0.702656
\(292\) −8.83679 −0.517134
\(293\) −4.47050 −0.261170 −0.130585 0.991437i \(-0.541685\pi\)
−0.130585 + 0.991437i \(0.541685\pi\)
\(294\) 4.76030 0.277627
\(295\) 0 0
\(296\) 1.61313 0.0937610
\(297\) −5.37300 −0.311773
\(298\) 1.63425 0.0946698
\(299\) −34.3246 −1.98504
\(300\) 0 0
\(301\) 6.34412 0.365669
\(302\) 2.14217 0.123268
\(303\) −13.8854 −0.797693
\(304\) 1.35163 0.0775215
\(305\) 0 0
\(306\) −0.449345 −0.0256873
\(307\) 25.1091 1.43305 0.716525 0.697562i \(-0.245732\pi\)
0.716525 + 0.697562i \(0.245732\pi\)
\(308\) −5.47960 −0.312229
\(309\) −13.6693 −0.777619
\(310\) 0 0
\(311\) −24.6576 −1.39821 −0.699103 0.715021i \(-0.746417\pi\)
−0.699103 + 0.715021i \(0.746417\pi\)
\(312\) −19.6459 −1.11223
\(313\) 14.7748 0.835123 0.417562 0.908649i \(-0.362885\pi\)
0.417562 + 0.908649i \(0.362885\pi\)
\(314\) 7.49136 0.422762
\(315\) 0 0
\(316\) 17.0050 0.956607
\(317\) −9.64687 −0.541822 −0.270911 0.962604i \(-0.587325\pi\)
−0.270911 + 0.962604i \(0.587325\pi\)
\(318\) 3.30181 0.185156
\(319\) −3.20619 −0.179512
\(320\) 0 0
\(321\) 10.5103 0.586626
\(322\) −12.1990 −0.679825
\(323\) 0.385149 0.0214303
\(324\) 1.68700 0.0937225
\(325\) 0 0
\(326\) 2.33472 0.129308
\(327\) 10.3900 0.574567
\(328\) −9.58086 −0.529015
\(329\) −7.95811 −0.438745
\(330\) 0 0
\(331\) −10.8051 −0.593903 −0.296952 0.954893i \(-0.595970\pi\)
−0.296952 + 0.954893i \(0.595970\pi\)
\(332\) 9.25132 0.507732
\(333\) 1.11436 0.0610667
\(334\) 5.80044 0.317386
\(335\) 0 0
\(336\) 5.58803 0.304852
\(337\) −35.0913 −1.91154 −0.955772 0.294110i \(-0.904977\pi\)
−0.955772 + 0.294110i \(0.904977\pi\)
\(338\) 25.1748 1.36933
\(339\) 0.802501 0.0435859
\(340\) 0 0
\(341\) 4.99714 0.270610
\(342\) −1.16668 −0.0630867
\(343\) −3.63382 −0.196208
\(344\) −4.29159 −0.231387
\(345\) 0 0
\(346\) 6.17758 0.332109
\(347\) 3.96060 0.212616 0.106308 0.994333i \(-0.466097\pi\)
0.106308 + 0.994333i \(0.466097\pi\)
\(348\) 5.59071 0.299693
\(349\) 16.1640 0.865241 0.432621 0.901576i \(-0.357589\pi\)
0.432621 + 0.901576i \(0.357589\pi\)
\(350\) 0 0
\(351\) −37.7436 −2.01460
\(352\) 5.81275 0.309821
\(353\) 20.9960 1.11751 0.558753 0.829334i \(-0.311280\pi\)
0.558753 + 0.829334i \(0.311280\pi\)
\(354\) 8.82231 0.468901
\(355\) 0 0
\(356\) 15.5162 0.822354
\(357\) 1.59231 0.0842741
\(358\) 5.78483 0.305738
\(359\) −27.1251 −1.43161 −0.715804 0.698301i \(-0.753940\pi\)
−0.715804 + 0.698301i \(0.753940\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 16.0273 0.842378
\(363\) 1.14700 0.0602020
\(364\) −38.4924 −2.01755
\(365\) 0 0
\(366\) 6.11908 0.319850
\(367\) −6.77494 −0.353649 −0.176824 0.984242i \(-0.556582\pi\)
−0.176824 + 0.984242i \(0.556582\pi\)
\(368\) −6.60449 −0.344283
\(369\) −6.61856 −0.344548
\(370\) 0 0
\(371\) 14.9800 0.777725
\(372\) −8.71364 −0.451781
\(373\) 2.57955 0.133564 0.0667821 0.997768i \(-0.478727\pi\)
0.0667821 + 0.997768i \(0.478727\pi\)
\(374\) 0.266771 0.0137944
\(375\) 0 0
\(376\) 5.38341 0.277628
\(377\) −22.5224 −1.15996
\(378\) −13.4141 −0.689948
\(379\) −8.19550 −0.420975 −0.210487 0.977597i \(-0.567505\pi\)
−0.210487 + 0.977597i \(0.567505\pi\)
\(380\) 0 0
\(381\) 21.3496 1.09377
\(382\) 7.94770 0.406640
\(383\) 36.7852 1.87963 0.939817 0.341677i \(-0.110995\pi\)
0.939817 + 0.341677i \(0.110995\pi\)
\(384\) −12.2835 −0.626839
\(385\) 0 0
\(386\) −12.0161 −0.611603
\(387\) −2.96468 −0.150703
\(388\) 15.8869 0.806534
\(389\) −20.6422 −1.04660 −0.523300 0.852149i \(-0.675299\pi\)
−0.523300 + 0.852149i \(0.675299\pi\)
\(390\) 0 0
\(391\) −1.88195 −0.0951744
\(392\) −14.6098 −0.737905
\(393\) −10.1204 −0.510507
\(394\) 9.42516 0.474832
\(395\) 0 0
\(396\) 2.56068 0.128679
\(397\) −15.3057 −0.768172 −0.384086 0.923297i \(-0.625483\pi\)
−0.384086 + 0.923297i \(0.625483\pi\)
\(398\) 6.06619 0.304071
\(399\) 4.13428 0.206973
\(400\) 0 0
\(401\) 26.2309 1.30991 0.654955 0.755668i \(-0.272687\pi\)
0.654955 + 0.755668i \(0.272687\pi\)
\(402\) −0.892024 −0.0444901
\(403\) 35.1033 1.74862
\(404\) 18.4037 0.915621
\(405\) 0 0
\(406\) −8.00450 −0.397257
\(407\) −0.661585 −0.0327935
\(408\) −1.07715 −0.0533268
\(409\) −4.93983 −0.244259 −0.122129 0.992514i \(-0.538972\pi\)
−0.122129 + 0.992514i \(0.538972\pi\)
\(410\) 0 0
\(411\) −3.56738 −0.175966
\(412\) 18.1174 0.892579
\(413\) 40.0261 1.96956
\(414\) 5.70074 0.280176
\(415\) 0 0
\(416\) 40.8326 2.00198
\(417\) 8.11878 0.397579
\(418\) 0.692644 0.0338783
\(419\) −8.64971 −0.422566 −0.211283 0.977425i \(-0.567764\pi\)
−0.211283 + 0.977425i \(0.567764\pi\)
\(420\) 0 0
\(421\) −29.7428 −1.44957 −0.724787 0.688973i \(-0.758062\pi\)
−0.724787 + 0.688973i \(0.758062\pi\)
\(422\) 14.0582 0.684342
\(423\) 3.71891 0.180820
\(424\) −10.1335 −0.492127
\(425\) 0 0
\(426\) 10.5889 0.513034
\(427\) 27.7618 1.34349
\(428\) −13.9304 −0.673350
\(429\) 8.05731 0.389011
\(430\) 0 0
\(431\) −32.4162 −1.56143 −0.780716 0.624885i \(-0.785146\pi\)
−0.780716 + 0.624885i \(0.785146\pi\)
\(432\) −7.26233 −0.349409
\(433\) 33.9587 1.63195 0.815975 0.578087i \(-0.196201\pi\)
0.815975 + 0.578087i \(0.196201\pi\)
\(434\) 12.4758 0.598855
\(435\) 0 0
\(436\) −13.7709 −0.659509
\(437\) −4.88630 −0.233743
\(438\) 4.61802 0.220657
\(439\) 14.4133 0.687907 0.343954 0.938987i \(-0.388234\pi\)
0.343954 + 0.938987i \(0.388234\pi\)
\(440\) 0 0
\(441\) −10.0926 −0.480599
\(442\) 1.87398 0.0891360
\(443\) −36.0311 −1.71189 −0.855943 0.517069i \(-0.827023\pi\)
−0.855943 + 0.517069i \(0.827023\pi\)
\(444\) 1.15362 0.0547485
\(445\) 0 0
\(446\) 13.7680 0.651933
\(447\) 2.70629 0.128003
\(448\) 4.76828 0.225280
\(449\) −10.3140 −0.486745 −0.243373 0.969933i \(-0.578254\pi\)
−0.243373 + 0.969933i \(0.578254\pi\)
\(450\) 0 0
\(451\) 3.92936 0.185026
\(452\) −1.06364 −0.0500294
\(453\) 3.54739 0.166671
\(454\) −0.362643 −0.0170197
\(455\) 0 0
\(456\) −2.79671 −0.130968
\(457\) −5.58176 −0.261104 −0.130552 0.991441i \(-0.541675\pi\)
−0.130552 + 0.991441i \(0.541675\pi\)
\(458\) 12.9334 0.604340
\(459\) −2.06941 −0.0965916
\(460\) 0 0
\(461\) 16.9193 0.788008 0.394004 0.919109i \(-0.371090\pi\)
0.394004 + 0.919109i \(0.371090\pi\)
\(462\) 2.86358 0.133226
\(463\) 0.827191 0.0384428 0.0192214 0.999815i \(-0.493881\pi\)
0.0192214 + 0.999815i \(0.493881\pi\)
\(464\) −4.33359 −0.201182
\(465\) 0 0
\(466\) −1.17688 −0.0545181
\(467\) −38.7142 −1.79148 −0.895739 0.444580i \(-0.853353\pi\)
−0.895739 + 0.444580i \(0.853353\pi\)
\(468\) 17.9879 0.831491
\(469\) −4.04704 −0.186875
\(470\) 0 0
\(471\) 12.4055 0.571616
\(472\) −27.0764 −1.24629
\(473\) 1.76009 0.0809292
\(474\) −8.88665 −0.408177
\(475\) 0 0
\(476\) −2.11046 −0.0967329
\(477\) −7.00034 −0.320523
\(478\) −20.5795 −0.941283
\(479\) 12.1245 0.553984 0.276992 0.960872i \(-0.410662\pi\)
0.276992 + 0.960872i \(0.410662\pi\)
\(480\) 0 0
\(481\) −4.64741 −0.211904
\(482\) −11.4998 −0.523804
\(483\) −20.2013 −0.919192
\(484\) −1.52024 −0.0691020
\(485\) 0 0
\(486\) 10.2831 0.466452
\(487\) −8.72457 −0.395348 −0.197674 0.980268i \(-0.563339\pi\)
−0.197674 + 0.980268i \(0.563339\pi\)
\(488\) −18.7800 −0.850129
\(489\) 3.86625 0.174838
\(490\) 0 0
\(491\) −42.1520 −1.90229 −0.951147 0.308737i \(-0.900094\pi\)
−0.951147 + 0.308737i \(0.900094\pi\)
\(492\) −6.85172 −0.308900
\(493\) −1.23486 −0.0556153
\(494\) 4.86559 0.218913
\(495\) 0 0
\(496\) 6.75431 0.303277
\(497\) 48.0410 2.15493
\(498\) −4.83464 −0.216645
\(499\) 18.9525 0.848431 0.424215 0.905561i \(-0.360550\pi\)
0.424215 + 0.905561i \(0.360550\pi\)
\(500\) 0 0
\(501\) 9.60539 0.429137
\(502\) 9.52291 0.425028
\(503\) −25.2153 −1.12430 −0.562148 0.827036i \(-0.690025\pi\)
−0.562148 + 0.827036i \(0.690025\pi\)
\(504\) 14.8033 0.659393
\(505\) 0 0
\(506\) −3.38446 −0.150458
\(507\) 41.6889 1.85147
\(508\) −28.2968 −1.25547
\(509\) −15.3133 −0.678751 −0.339375 0.940651i \(-0.610216\pi\)
−0.339375 + 0.940651i \(0.610216\pi\)
\(510\) 0 0
\(511\) 20.9516 0.926843
\(512\) 14.4480 0.638518
\(513\) −5.37300 −0.237224
\(514\) 18.5602 0.818657
\(515\) 0 0
\(516\) −3.06912 −0.135110
\(517\) −2.20788 −0.0971022
\(518\) −1.65170 −0.0725714
\(519\) 10.2299 0.449045
\(520\) 0 0
\(521\) −6.62604 −0.290292 −0.145146 0.989410i \(-0.546365\pi\)
−0.145146 + 0.989410i \(0.546365\pi\)
\(522\) 3.74059 0.163721
\(523\) −10.2228 −0.447014 −0.223507 0.974702i \(-0.571750\pi\)
−0.223507 + 0.974702i \(0.571750\pi\)
\(524\) 13.4136 0.585978
\(525\) 0 0
\(526\) −11.9381 −0.520526
\(527\) 1.92464 0.0838388
\(528\) 1.55033 0.0674694
\(529\) 0.875897 0.0380825
\(530\) 0 0
\(531\) −18.7047 −0.811713
\(532\) −5.47960 −0.237571
\(533\) 27.6025 1.19560
\(534\) −8.10858 −0.350893
\(535\) 0 0
\(536\) 2.73770 0.118250
\(537\) 9.57954 0.413388
\(538\) −12.4200 −0.535465
\(539\) 5.99184 0.258087
\(540\) 0 0
\(541\) −11.5547 −0.496777 −0.248388 0.968661i \(-0.579901\pi\)
−0.248388 + 0.968661i \(0.579901\pi\)
\(542\) −6.75440 −0.290126
\(543\) 26.5409 1.13898
\(544\) 2.23877 0.0959867
\(545\) 0 0
\(546\) 20.1157 0.860873
\(547\) −2.09444 −0.0895518 −0.0447759 0.998997i \(-0.514257\pi\)
−0.0447759 + 0.998997i \(0.514257\pi\)
\(548\) 4.72823 0.201980
\(549\) −12.9734 −0.553691
\(550\) 0 0
\(551\) −3.20619 −0.136588
\(552\) 13.6655 0.581644
\(553\) −40.3180 −1.71450
\(554\) −14.4867 −0.615482
\(555\) 0 0
\(556\) −10.7607 −0.456355
\(557\) 4.21753 0.178703 0.0893513 0.996000i \(-0.471521\pi\)
0.0893513 + 0.996000i \(0.471521\pi\)
\(558\) −5.83006 −0.246806
\(559\) 12.3641 0.522945
\(560\) 0 0
\(561\) 0.441767 0.0186514
\(562\) 1.05904 0.0446729
\(563\) 8.55912 0.360724 0.180362 0.983600i \(-0.442273\pi\)
0.180362 + 0.983600i \(0.442273\pi\)
\(564\) 3.84993 0.162111
\(565\) 0 0
\(566\) 5.49038 0.230778
\(567\) −3.99980 −0.167976
\(568\) −32.4982 −1.36359
\(569\) 8.10736 0.339878 0.169939 0.985455i \(-0.445643\pi\)
0.169939 + 0.985455i \(0.445643\pi\)
\(570\) 0 0
\(571\) 8.88932 0.372006 0.186003 0.982549i \(-0.440447\pi\)
0.186003 + 0.982549i \(0.440447\pi\)
\(572\) −10.6792 −0.446520
\(573\) 13.1612 0.549818
\(574\) 9.80996 0.409460
\(575\) 0 0
\(576\) −2.22827 −0.0928445
\(577\) 26.4638 1.10170 0.550852 0.834603i \(-0.314303\pi\)
0.550852 + 0.834603i \(0.314303\pi\)
\(578\) −11.6722 −0.485499
\(579\) −19.8984 −0.826949
\(580\) 0 0
\(581\) −21.9344 −0.909992
\(582\) −8.30232 −0.344142
\(583\) 4.15602 0.172125
\(584\) −14.1731 −0.586486
\(585\) 0 0
\(586\) −3.09647 −0.127914
\(587\) −11.2994 −0.466376 −0.233188 0.972432i \(-0.574916\pi\)
−0.233188 + 0.972432i \(0.574916\pi\)
\(588\) −10.4481 −0.430874
\(589\) 4.99714 0.205904
\(590\) 0 0
\(591\) 15.6079 0.642021
\(592\) −0.894220 −0.0367522
\(593\) −47.7149 −1.95942 −0.979708 0.200432i \(-0.935765\pi\)
−0.979708 + 0.200432i \(0.935765\pi\)
\(594\) −3.72158 −0.152698
\(595\) 0 0
\(596\) −3.58693 −0.146927
\(597\) 10.0455 0.411134
\(598\) −23.7747 −0.972221
\(599\) −3.68890 −0.150724 −0.0753622 0.997156i \(-0.524011\pi\)
−0.0753622 + 0.997156i \(0.524011\pi\)
\(600\) 0 0
\(601\) −20.9917 −0.856268 −0.428134 0.903715i \(-0.640829\pi\)
−0.428134 + 0.903715i \(0.640829\pi\)
\(602\) 4.39421 0.179095
\(603\) 1.89123 0.0770167
\(604\) −4.70174 −0.191311
\(605\) 0 0
\(606\) −9.61761 −0.390689
\(607\) 33.2697 1.35037 0.675187 0.737647i \(-0.264063\pi\)
0.675187 + 0.737647i \(0.264063\pi\)
\(608\) 5.81275 0.235738
\(609\) −13.2553 −0.537131
\(610\) 0 0
\(611\) −15.5096 −0.627451
\(612\) 0.986242 0.0398665
\(613\) 39.8232 1.60844 0.804222 0.594330i \(-0.202583\pi\)
0.804222 + 0.594330i \(0.202583\pi\)
\(614\) 17.3916 0.701869
\(615\) 0 0
\(616\) −8.78857 −0.354101
\(617\) 23.3355 0.939454 0.469727 0.882812i \(-0.344352\pi\)
0.469727 + 0.882812i \(0.344352\pi\)
\(618\) −9.46795 −0.380857
\(619\) −18.4550 −0.741771 −0.370886 0.928679i \(-0.620946\pi\)
−0.370886 + 0.928679i \(0.620946\pi\)
\(620\) 0 0
\(621\) 26.2541 1.05354
\(622\) −17.0790 −0.684804
\(623\) −36.7880 −1.47388
\(624\) 10.8905 0.435970
\(625\) 0 0
\(626\) 10.2337 0.409021
\(627\) 1.14700 0.0458069
\(628\) −16.4424 −0.656122
\(629\) −0.254809 −0.0101599
\(630\) 0 0
\(631\) −29.2519 −1.16450 −0.582249 0.813010i \(-0.697827\pi\)
−0.582249 + 0.813010i \(0.697827\pi\)
\(632\) 27.2738 1.08490
\(633\) 23.2800 0.925299
\(634\) −6.68184 −0.265370
\(635\) 0 0
\(636\) −7.24695 −0.287360
\(637\) 42.0907 1.66769
\(638\) −2.22074 −0.0879201
\(639\) −22.4501 −0.888112
\(640\) 0 0
\(641\) 25.9437 1.02471 0.512357 0.858772i \(-0.328772\pi\)
0.512357 + 0.858772i \(0.328772\pi\)
\(642\) 7.27987 0.287313
\(643\) 41.8049 1.64863 0.824313 0.566135i \(-0.191562\pi\)
0.824313 + 0.566135i \(0.191562\pi\)
\(644\) 26.7750 1.05508
\(645\) 0 0
\(646\) 0.266771 0.0104960
\(647\) −17.2410 −0.677815 −0.338907 0.940820i \(-0.610057\pi\)
−0.338907 + 0.940820i \(0.610057\pi\)
\(648\) 2.70574 0.106291
\(649\) 11.1047 0.435899
\(650\) 0 0
\(651\) 20.6596 0.809713
\(652\) −5.12436 −0.200685
\(653\) −23.0285 −0.901174 −0.450587 0.892733i \(-0.648785\pi\)
−0.450587 + 0.892733i \(0.648785\pi\)
\(654\) 7.19655 0.281408
\(655\) 0 0
\(656\) 5.31106 0.207362
\(657\) −9.79090 −0.381979
\(658\) −5.51213 −0.214885
\(659\) 43.1431 1.68062 0.840309 0.542108i \(-0.182374\pi\)
0.840309 + 0.542108i \(0.182374\pi\)
\(660\) 0 0
\(661\) −23.0871 −0.897985 −0.448993 0.893535i \(-0.648217\pi\)
−0.448993 + 0.893535i \(0.648217\pi\)
\(662\) −7.48410 −0.290878
\(663\) 3.10326 0.120521
\(664\) 14.8379 0.575823
\(665\) 0 0
\(666\) 0.771856 0.0299088
\(667\) 15.6664 0.606605
\(668\) −12.7311 −0.492579
\(669\) 22.7995 0.881479
\(670\) 0 0
\(671\) 7.70215 0.297338
\(672\) 24.0315 0.927037
\(673\) −42.0990 −1.62280 −0.811398 0.584494i \(-0.801293\pi\)
−0.811398 + 0.584494i \(0.801293\pi\)
\(674\) −24.3057 −0.936222
\(675\) 0 0
\(676\) −55.2547 −2.12518
\(677\) −51.2657 −1.97030 −0.985151 0.171693i \(-0.945076\pi\)
−0.985151 + 0.171693i \(0.945076\pi\)
\(678\) 0.555847 0.0213472
\(679\) −37.6670 −1.44552
\(680\) 0 0
\(681\) −0.600529 −0.0230123
\(682\) 3.46124 0.132538
\(683\) 21.0036 0.803680 0.401840 0.915710i \(-0.368371\pi\)
0.401840 + 0.915710i \(0.368371\pi\)
\(684\) 2.56068 0.0979099
\(685\) 0 0
\(686\) −2.51694 −0.0960972
\(687\) 21.4175 0.817129
\(688\) 2.37900 0.0906987
\(689\) 29.1947 1.11223
\(690\) 0 0
\(691\) −43.6608 −1.66094 −0.830468 0.557066i \(-0.811927\pi\)
−0.830468 + 0.557066i \(0.811927\pi\)
\(692\) −13.5588 −0.515430
\(693\) −6.07123 −0.230627
\(694\) 2.74329 0.104134
\(695\) 0 0
\(696\) 8.96677 0.339884
\(697\) 1.51339 0.0573237
\(698\) 11.1959 0.423772
\(699\) −1.94889 −0.0737139
\(700\) 0 0
\(701\) 1.36357 0.0515013 0.0257506 0.999668i \(-0.491802\pi\)
0.0257506 + 0.999668i \(0.491802\pi\)
\(702\) −26.1428 −0.986698
\(703\) −0.661585 −0.0249521
\(704\) 1.32290 0.0498585
\(705\) 0 0
\(706\) 14.5428 0.547324
\(707\) −43.6343 −1.64104
\(708\) −19.3636 −0.727729
\(709\) 24.2648 0.911285 0.455642 0.890163i \(-0.349410\pi\)
0.455642 + 0.890163i \(0.349410\pi\)
\(710\) 0 0
\(711\) 18.8410 0.706594
\(712\) 24.8859 0.932638
\(713\) −24.4175 −0.914443
\(714\) 1.10291 0.0412752
\(715\) 0 0
\(716\) −12.6968 −0.474501
\(717\) −34.0792 −1.27271
\(718\) −18.7880 −0.701163
\(719\) 18.2569 0.680867 0.340434 0.940269i \(-0.389426\pi\)
0.340434 + 0.940269i \(0.389426\pi\)
\(720\) 0 0
\(721\) −42.9553 −1.59974
\(722\) 0.692644 0.0257775
\(723\) −19.0435 −0.708235
\(724\) −35.1775 −1.30736
\(725\) 0 0
\(726\) 0.794464 0.0294853
\(727\) −34.8231 −1.29152 −0.645759 0.763541i \(-0.723459\pi\)
−0.645759 + 0.763541i \(0.723459\pi\)
\(728\) −61.7368 −2.28812
\(729\) 20.3577 0.753989
\(730\) 0 0
\(731\) 0.677898 0.0250730
\(732\) −13.4304 −0.496403
\(733\) −15.1353 −0.559036 −0.279518 0.960141i \(-0.590175\pi\)
−0.279518 + 0.960141i \(0.590175\pi\)
\(734\) −4.69262 −0.173208
\(735\) 0 0
\(736\) −28.4028 −1.04694
\(737\) −1.12280 −0.0413589
\(738\) −4.58430 −0.168750
\(739\) −25.7057 −0.945600 −0.472800 0.881170i \(-0.656757\pi\)
−0.472800 + 0.881170i \(0.656757\pi\)
\(740\) 0 0
\(741\) 8.05731 0.295993
\(742\) 10.3758 0.380909
\(743\) −25.9533 −0.952136 −0.476068 0.879409i \(-0.657938\pi\)
−0.476068 + 0.879409i \(0.657938\pi\)
\(744\) −13.9755 −0.512368
\(745\) 0 0
\(746\) 1.78671 0.0654162
\(747\) 10.2502 0.375034
\(748\) −0.585521 −0.0214088
\(749\) 33.0282 1.20682
\(750\) 0 0
\(751\) −26.5830 −0.970026 −0.485013 0.874507i \(-0.661185\pi\)
−0.485013 + 0.874507i \(0.661185\pi\)
\(752\) −2.98424 −0.108824
\(753\) 15.7697 0.574681
\(754\) −15.6000 −0.568118
\(755\) 0 0
\(756\) 29.4419 1.07079
\(757\) −0.372319 −0.0135322 −0.00676608 0.999977i \(-0.502154\pi\)
−0.00676608 + 0.999977i \(0.502154\pi\)
\(758\) −5.67656 −0.206182
\(759\) −5.60460 −0.203434
\(760\) 0 0
\(761\) −12.7607 −0.462574 −0.231287 0.972886i \(-0.574294\pi\)
−0.231287 + 0.972886i \(0.574294\pi\)
\(762\) 14.7876 0.535699
\(763\) 32.6502 1.18202
\(764\) −17.4440 −0.631100
\(765\) 0 0
\(766\) 25.4790 0.920594
\(767\) 78.0071 2.81667
\(768\) −11.5428 −0.416515
\(769\) 46.2851 1.66908 0.834542 0.550944i \(-0.185732\pi\)
0.834542 + 0.550944i \(0.185732\pi\)
\(770\) 0 0
\(771\) 30.7354 1.10691
\(772\) 26.3735 0.949202
\(773\) −12.5661 −0.451973 −0.225986 0.974130i \(-0.572561\pi\)
−0.225986 + 0.974130i \(0.572561\pi\)
\(774\) −2.05346 −0.0738102
\(775\) 0 0
\(776\) 25.4805 0.914696
\(777\) −2.73518 −0.0981239
\(778\) −14.2977 −0.512596
\(779\) 3.92936 0.140784
\(780\) 0 0
\(781\) 13.3284 0.476926
\(782\) −1.30352 −0.0466138
\(783\) 17.2268 0.615637
\(784\) 8.09878 0.289242
\(785\) 0 0
\(786\) −7.00983 −0.250032
\(787\) −14.6706 −0.522952 −0.261476 0.965210i \(-0.584209\pi\)
−0.261476 + 0.965210i \(0.584209\pi\)
\(788\) −20.6867 −0.736935
\(789\) −19.7692 −0.703803
\(790\) 0 0
\(791\) 2.52183 0.0896661
\(792\) 4.10699 0.145936
\(793\) 54.1051 1.92133
\(794\) −10.6014 −0.376230
\(795\) 0 0
\(796\) −13.3144 −0.471915
\(797\) −37.6027 −1.33196 −0.665978 0.745972i \(-0.731985\pi\)
−0.665978 + 0.745972i \(0.731985\pi\)
\(798\) 2.86358 0.101370
\(799\) −0.850361 −0.0300836
\(800\) 0 0
\(801\) 17.1914 0.607429
\(802\) 18.1687 0.641559
\(803\) 5.81274 0.205127
\(804\) 1.95785 0.0690482
\(805\) 0 0
\(806\) 24.3140 0.856426
\(807\) −20.5673 −0.724002
\(808\) 29.5172 1.03841
\(809\) 5.32635 0.187264 0.0936322 0.995607i \(-0.470152\pi\)
0.0936322 + 0.995607i \(0.470152\pi\)
\(810\) 0 0
\(811\) 1.82625 0.0641282 0.0320641 0.999486i \(-0.489792\pi\)
0.0320641 + 0.999486i \(0.489792\pi\)
\(812\) 17.5686 0.616538
\(813\) −11.1851 −0.392280
\(814\) −0.458242 −0.0160614
\(815\) 0 0
\(816\) 0.597107 0.0209029
\(817\) 1.76009 0.0615779
\(818\) −3.42154 −0.119631
\(819\) −42.6484 −1.49025
\(820\) 0 0
\(821\) −54.3594 −1.89716 −0.948579 0.316541i \(-0.897479\pi\)
−0.948579 + 0.316541i \(0.897479\pi\)
\(822\) −2.47092 −0.0861833
\(823\) −46.5557 −1.62283 −0.811415 0.584471i \(-0.801302\pi\)
−0.811415 + 0.584471i \(0.801302\pi\)
\(824\) 29.0579 1.01228
\(825\) 0 0
\(826\) 27.7238 0.964636
\(827\) 1.34120 0.0466380 0.0233190 0.999728i \(-0.492577\pi\)
0.0233190 + 0.999728i \(0.492577\pi\)
\(828\) −12.5122 −0.434830
\(829\) −42.8307 −1.48757 −0.743785 0.668419i \(-0.766972\pi\)
−0.743785 + 0.668419i \(0.766972\pi\)
\(830\) 0 0
\(831\) −23.9897 −0.832194
\(832\) 9.29291 0.322174
\(833\) 2.30775 0.0799589
\(834\) 5.62342 0.194723
\(835\) 0 0
\(836\) −1.52024 −0.0525788
\(837\) −26.8496 −0.928060
\(838\) −5.99117 −0.206961
\(839\) 8.45704 0.291970 0.145985 0.989287i \(-0.453365\pi\)
0.145985 + 0.989287i \(0.453365\pi\)
\(840\) 0 0
\(841\) −18.7204 −0.645530
\(842\) −20.6011 −0.709962
\(843\) 1.75375 0.0604023
\(844\) −30.8555 −1.06209
\(845\) 0 0
\(846\) 2.57588 0.0885606
\(847\) 3.60442 0.123849
\(848\) 5.61742 0.192903
\(849\) 9.09196 0.312035
\(850\) 0 0
\(851\) 3.23270 0.110815
\(852\) −23.2410 −0.796223
\(853\) 2.07765 0.0711372 0.0355686 0.999367i \(-0.488676\pi\)
0.0355686 + 0.999367i \(0.488676\pi\)
\(854\) 19.2290 0.658004
\(855\) 0 0
\(856\) −22.3425 −0.763651
\(857\) −10.3114 −0.352230 −0.176115 0.984370i \(-0.556353\pi\)
−0.176115 + 0.984370i \(0.556353\pi\)
\(858\) 5.58085 0.190527
\(859\) 8.80362 0.300376 0.150188 0.988657i \(-0.452012\pi\)
0.150188 + 0.988657i \(0.452012\pi\)
\(860\) 0 0
\(861\) 16.2451 0.553631
\(862\) −22.4529 −0.764748
\(863\) −7.62460 −0.259544 −0.129772 0.991544i \(-0.541425\pi\)
−0.129772 + 0.991544i \(0.541425\pi\)
\(864\) −31.2319 −1.06253
\(865\) 0 0
\(866\) 23.5213 0.799285
\(867\) −19.3289 −0.656444
\(868\) −27.3823 −0.929417
\(869\) −11.1857 −0.379449
\(870\) 0 0
\(871\) −7.88730 −0.267251
\(872\) −22.0868 −0.747954
\(873\) 17.6022 0.595743
\(874\) −3.38446 −0.114481
\(875\) 0 0
\(876\) −10.1358 −0.342458
\(877\) −34.3933 −1.16138 −0.580689 0.814125i \(-0.697217\pi\)
−0.580689 + 0.814125i \(0.697217\pi\)
\(878\) 9.98325 0.336919
\(879\) −5.12768 −0.172952
\(880\) 0 0
\(881\) 8.09345 0.272675 0.136338 0.990662i \(-0.456467\pi\)
0.136338 + 0.990662i \(0.456467\pi\)
\(882\) −6.99056 −0.235384
\(883\) 26.3719 0.887486 0.443743 0.896154i \(-0.353650\pi\)
0.443743 + 0.896154i \(0.353650\pi\)
\(884\) −4.11309 −0.138338
\(885\) 0 0
\(886\) −24.9567 −0.838436
\(887\) −35.2984 −1.18520 −0.592602 0.805496i \(-0.701899\pi\)
−0.592602 + 0.805496i \(0.701899\pi\)
\(888\) 1.85026 0.0620906
\(889\) 67.0903 2.25014
\(890\) 0 0
\(891\) −1.10969 −0.0371761
\(892\) −30.2186 −1.01179
\(893\) −2.20788 −0.0738837
\(894\) 1.87449 0.0626925
\(895\) 0 0
\(896\) −38.6005 −1.28955
\(897\) −39.3704 −1.31454
\(898\) −7.14389 −0.238395
\(899\) −16.0218 −0.534356
\(900\) 0 0
\(901\) 1.60069 0.0533266
\(902\) 2.72165 0.0906209
\(903\) 7.27672 0.242154
\(904\) −1.70594 −0.0567387
\(905\) 0 0
\(906\) 2.45708 0.0816310
\(907\) −57.0934 −1.89576 −0.947878 0.318634i \(-0.896776\pi\)
−0.947878 + 0.318634i \(0.896776\pi\)
\(908\) 0.795945 0.0264144
\(909\) 20.3908 0.676320
\(910\) 0 0
\(911\) −18.2973 −0.606216 −0.303108 0.952956i \(-0.598024\pi\)
−0.303108 + 0.952956i \(0.598024\pi\)
\(912\) 1.55033 0.0513365
\(913\) −6.08541 −0.201398
\(914\) −3.86617 −0.127882
\(915\) 0 0
\(916\) −28.3869 −0.937930
\(917\) −31.8030 −1.05023
\(918\) −1.43336 −0.0473079
\(919\) −53.1018 −1.75167 −0.875833 0.482614i \(-0.839688\pi\)
−0.875833 + 0.482614i \(0.839688\pi\)
\(920\) 0 0
\(921\) 28.8001 0.948997
\(922\) 11.7190 0.385945
\(923\) 93.6273 3.08178
\(924\) −6.28512 −0.206765
\(925\) 0 0
\(926\) 0.572948 0.0188282
\(927\) 20.0735 0.659300
\(928\) −18.6368 −0.611782
\(929\) 28.2532 0.926957 0.463479 0.886108i \(-0.346601\pi\)
0.463479 + 0.886108i \(0.346601\pi\)
\(930\) 0 0
\(931\) 5.99184 0.196375
\(932\) 2.58308 0.0846114
\(933\) −28.2824 −0.925923
\(934\) −26.8151 −0.877418
\(935\) 0 0
\(936\) 28.8503 0.943000
\(937\) −15.6921 −0.512637 −0.256319 0.966592i \(-0.582510\pi\)
−0.256319 + 0.966592i \(0.582510\pi\)
\(938\) −2.80316 −0.0915264
\(939\) 16.9468 0.553037
\(940\) 0 0
\(941\) 16.4701 0.536910 0.268455 0.963292i \(-0.413487\pi\)
0.268455 + 0.963292i \(0.413487\pi\)
\(942\) 8.59260 0.279962
\(943\) −19.2000 −0.625239
\(944\) 15.0095 0.488519
\(945\) 0 0
\(946\) 1.21912 0.0396370
\(947\) 26.2431 0.852785 0.426392 0.904538i \(-0.359784\pi\)
0.426392 + 0.904538i \(0.359784\pi\)
\(948\) 19.5048 0.633487
\(949\) 40.8326 1.32548
\(950\) 0 0
\(951\) −11.0650 −0.358807
\(952\) −3.38491 −0.109705
\(953\) −37.0207 −1.19922 −0.599610 0.800293i \(-0.704677\pi\)
−0.599610 + 0.800293i \(0.704677\pi\)
\(954\) −4.84874 −0.156984
\(955\) 0 0
\(956\) 45.1687 1.46086
\(957\) −3.67750 −0.118877
\(958\) 8.39798 0.271326
\(959\) −11.2104 −0.362002
\(960\) 0 0
\(961\) −6.02858 −0.194470
\(962\) −3.21900 −0.103785
\(963\) −15.4344 −0.497367
\(964\) 25.2404 0.812938
\(965\) 0 0
\(966\) −13.9923 −0.450195
\(967\) 52.9984 1.70431 0.852157 0.523286i \(-0.175294\pi\)
0.852157 + 0.523286i \(0.175294\pi\)
\(968\) −2.43828 −0.0783691
\(969\) 0.441767 0.0141916
\(970\) 0 0
\(971\) 16.5465 0.531003 0.265501 0.964110i \(-0.414463\pi\)
0.265501 + 0.964110i \(0.414463\pi\)
\(972\) −22.5698 −0.723928
\(973\) 25.5130 0.817910
\(974\) −6.04302 −0.193631
\(975\) 0 0
\(976\) 10.4105 0.333232
\(977\) −7.03173 −0.224965 −0.112482 0.993654i \(-0.535880\pi\)
−0.112482 + 0.993654i \(0.535880\pi\)
\(978\) 2.67793 0.0856309
\(979\) −10.2064 −0.326196
\(980\) 0 0
\(981\) −15.2578 −0.487144
\(982\) −29.1963 −0.931693
\(983\) −5.31776 −0.169610 −0.0848052 0.996398i \(-0.527027\pi\)
−0.0848052 + 0.996398i \(0.527027\pi\)
\(984\) −10.9893 −0.350325
\(985\) 0 0
\(986\) −0.855317 −0.0272389
\(987\) −9.12797 −0.290547
\(988\) −10.6792 −0.339751
\(989\) −8.60034 −0.273475
\(990\) 0 0
\(991\) −33.9658 −1.07896 −0.539480 0.841998i \(-0.681379\pi\)
−0.539480 + 0.841998i \(0.681379\pi\)
\(992\) 29.0471 0.922247
\(993\) −12.3935 −0.393296
\(994\) 33.2753 1.05543
\(995\) 0 0
\(996\) 10.6113 0.336231
\(997\) 48.4773 1.53529 0.767645 0.640875i \(-0.221428\pi\)
0.767645 + 0.640875i \(0.221428\pi\)
\(998\) 13.1273 0.415538
\(999\) 3.55470 0.112466
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5225.2.a.y.1.8 yes 15
5.4 even 2 5225.2.a.r.1.8 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.r.1.8 15 5.4 even 2
5225.2.a.y.1.8 yes 15 1.1 even 1 trivial